The Effect of Fiber Reinforcement on Crack Development due to Restraint of Concrete Structures

Crack Development due to Restraint of Concrete Structures

Elisabeth Leite Skare

Civil and Environmental Engineering Supervisor: Terje Kanstad, KT

Co-supervisor: Kathrin Sandstad, Statens vegvesen Håvard Johansen, Statens vegvesen

Department of Structural Engineering Submission date: June 2016

Norwegian University of Science and Technology

Faculty of Engineering Science and Technology

NTNU- Norwegian University of Science and Technology

MASTER THESIS 2016

SUBJECT AREA:

Concrete Technology

DATE:

09.06.16

NO. OF PAGES:

16 + 82 +10

TITLE:

The Effect of Fiber Reinforcement on Crack Development due to Restraint of Concrete Structures

BY:

Elisabeth Leite Skare

RESPONSIBLE TEACHER: Professor Terje Kanstad

SUPERVISOR: Professor Terje Kanstad

CARRIED OUT AT: The Department of Structural Engineering, NTNU SUMMARY:

This thesis studies the effect of fiber reinforcement on crack development in concrete structures. Background for choice of subject, is that cracks due to external restraint are observed on many of today's bridges, and it is assumed that fiber reinforcement may limit this crack development. Hence, a laboratory experiment was performed during the summer of 2015, in connection with the specialization project written by the

undersigned. The experiment studied the effect of steel fibers and polymer fibers on the crack development, compared to a concrete without fiber reinforcement. In the fiber mixes, one saw a much denser crack development than for the reference concrete, and the crack widths were much smaller in the fiber mixes.

On the initiative of Norwegian Public Roads Administration, it was decided to carry out the same measures to reduce cracks on a real bridge. This full-scale experiment has been conducted in connection with this thesis.

The chosen bridge is the pedestrian and bicycle bridge Sandsgård Bridge in Ganddal, close to Sandnes. The bridge is 50 meters long and the edge beams, á 16 meters, were cast with six different concrete mixes, five of them containing fiber reinforcement.

First some relevant theoretical background is presented, and three models for calculation crack widths are introduced. Then a brief description of the laboratory experiment is presented, followed by a description of the full-scale experiment on Sandsgård Bridge. Calculations of the crack widths are performed for the two experiments.

Simulations of the two experiments are performed in the finite element program, CrackTeStCOIN. This computer program is used for simulation of the temperature and stress development of the experiments.

Based on the simulated stresses, and an assumption of an effective elastic modulus of 10 000 MPa and 12 000 MPa for the laboratory and full-scale experiment respectively, the occurring strains are calculated. The simulated strains are compared with the strains obtained from the three investigated calculation models.

Lastly, the effect of the different fiber reinforcement are discussed.

OPEN

Fakultet for ingeniørvitenskap og teknologi

NTNU- Norges teknisk- naturvitenskapelige universitet

MASTEROPPGAVE 2016

FAGOMRÅDE:

Betongteknologi

DATO:

09.06.16

ANTALL SIDER:

16 + 82 + 10

TITTEL:

Virkning av fiberarmering på fastholdningsriss i betongkonstruksjoner

UTFØRT AV:

Elisabeth Leite Skare

FAGLÆRER: Professor Terje Kanstad

VEILEDER: Professor Terje Kanstad

UTFØRT VED: Institutt for konstruksjonsteknikk, NTNU SAMMENDRAG:

Denne oppgaven tar for seg virkningen av fiberarmering på rissutvikling i betongkonstruksjoner. Bakgrunn for oppgavens tema, er at det er observert fastholdningsriss på kantdragerne på mange av dagens broer. Det antas at fiberarmering kan begrense slik rissutvikling, og det ble derfor utført et laboratorieforsøk, sommeren 2015, i forbindelse med fordypningsprosjektet til undertegnede. Forsøket studerte effekten av stålfiber og polymerfiber på rissutvikling, sammenlignet med rissutviklingen i en betongbjelke uten fiberarmering. I fiberbetongene ble det observert en mye tettere rissopptreden enn i referansebetongen, og for fiberbetongene var rissviddene mindre enn for referansebetongen.

Etter initiativ fra Statens Vegvesen ble det bestemt å utføre de samme rissreduserende tiltakene på en ekte bro. Dette fullskala-forsøket er blitt utført i forbindelse med denne oppgaven. Den valgte broen er gang- og sykkelbroen Sandsgård Bro i Ganddal, i nærheten av Sandnes. Broen er 50 meter lang, og kantdragerne á 16 meter, ble støpt med seks ulike betongblandinger, hvor av fem inneholdt fiberarmering.

Først blir relevant bakgrunnsteori presentert, og tre beregningsmodeller for rissvidder blir introdusert.

Deretter blir en kort beskrivelse av laboratorieforsøket gitt, etterfulgt av en beskrivelse av fullskala-forsøket på Sandsgård Bro. Rissvidde-beregninger etter beregningsmodellene er utført for begge forsøkene.

Simuleringer av de to forsøkene er utført i elementprogrammet CrackTeStCOIN. Dette dataprogrammet blir brukt til å estimere temperatur- og spenningsutviklingen til de to forsøkene. Basert på de simulerte

spenningene, og en antagelse om en effektiv E-modul lik henholdsvis 10 000 MPa og 12 000 MPa for laboratorie- og fullskalaforsøket, beregnes de opptredende tøyningene. Disse blir sammenlignet med de beregnede tøyningene fra beregningsmodellene.

Til slutt blir effekten av de ulike fiberarmeringene drøftet.

ÅPEN

This Masters thesis was written at the Department of Structural Engineering at the Norwegian University of Science and Technology in Trondheim, during the autumn 2016. The thesis is a requirement for the degree of Master of Science, and finalized my study in the programme Civil and Environmental Engineering, with specialization in Structural Engineering.

First, I would like to thank Researcher Giedrius Zirgulis at NTNU for conducting the residual flexural bending strength tests, as well as providing assistance with the experimental set up. I will also like to thank Engineer Steinar Seehuus, Staff Engineer Ove Loraas and Staff Engineer Gøran Loraas for good guidance at the laboratory and for providing the required equipment, both in conjunction with this thesis and the specialization project.

A special thanks is given to Nils Bergfinn Haaland at Skanska, for your warm welcome at Sandsg˚ard Bridge, and for providing required equipment at the construction site, as well as useful information by mail. Also, Stian Persson, Kathrin Sandstad and Sascha Baarck at the Norwegian Public Roads Administration, have been helpful with providing the necessary draw- ings and information by mail, and supervision on the construction site. Concrete technologist Arne Vatnar at Skanska Teknikk has been helpful with providing material information by mail.

He has installed the heating cables at Sandsg˚ard Bridge, and processed the temperature data in excel. He also has participated in casting of the edge beams, and the prisms for testing of the residual flexural tensile strength.

I would also like thank Anja Birgitta Estenstad Klausen for help regarding the finite element program CrackTeStCOIN.

Finally, I would like to thank my main supervisor, Professor Terje Kanstad; first of all for good guidance, supervision and feedback. He has shown great interest in the topic, as evidenced by the fact that he travelled to Sandnes with me on a construction site visit. Secondly, I would like to thank him for contributing with relevant background theory and his eager for follow-up meetings. He has been willingly answering my questions and helping with the proofreading.

Trondheim, June 9, 2016

v

This thesis studies the effect of fiber reinforcement on crack development in concrete structures.

Background for choice of subject, is that cracks due to external restraint are observed on many of today’s bridges, and it is assumed that fiber reinforcement may limit this crack development.

Hence, a laboratory experiment was performed during the summer of 2015, in connection with the specialization project written by the undersigned. The experiment studied the effect of steel fibers and polymer fibers on the crack development, compared to a concrete without fiber reinforcement. In the fiber mixes, one saw a much denser crack development than for the reference concrete, and the crack widths were much smaller in the fiber mixes.

On the initiative of Norwegian Public Roads Administration, it was decided to carry out the same measures to reduce cracks on a real bridge. This full-scale experiment has been conducted in connection with this thesis. The chosen bridge is the pedestrian and bicycle bridge Sandsg˚ard Bridge in Ganddal, close to Sandnes. The bridge is 50 meters long and the edge beams, `a 16 meters, were cast with six different concrete mixes, five of them containing fiber reinforcement.

First some relevant theoretical background is presented, and three models for calculation crack widths are introduced. Then a brief description of the laboratory experiment is presented, followed by a description of the full-scale experiment on Sandsg˚ard Bridge. Calculations of the crack widths are performed for the two experiments.

Simulations of the two experiments are performed in the finite element program, CrackTeSt- COIN. This computer program is used for simulation of the temperature and stress development of the experiments. Based on the simulated stresses, and an assumption of an effective elastic modulus of 10 000 MPa and 12 000 MPa for the laboratory and full-scale experiment respec- tively, the occurring strains are calculated. The simulated strains are compared with the strains obtained from the three investigated calculation models.

Lastly, the effect of the different fiber reinforcement are discussed.

vii

Preface v

Abstract vii

1 Introduction 1

1.1 Background . . . 1

2 Theory 5 2.1 General . . . 5

2.2 Temperature . . . 5

2.3 Effect of Temperature on the Material Properties of Concrete . . . 7

2.3.1 General . . . 7

2.3.2 Material Properties at Sub-zero Temperatures . . . 7

2.3.3 Material Properties at Elevated Temperatures . . . 7

2.4 Creep and Shrinkage of Concrete . . . 8

2.5 Autogenous Shrinkage . . . 8

2.6 Strains and Stresses in Concrete Sections Subjected to Restrained Imposed De- formations . . . 9

2.7 Calculation of Crack Widths due to Restraint of Imposed Deformations . . . 10

2.7.1 General . . . 10

2.7.2 Restraint of a Member at its End . . . 10

2.7.3 Restraint Along One Edge . . . 11

2.8 Measures to Reduce Effects of Cracking . . . 11

2.9 Early-age Thermal Crack Control in Concrete . . . 13

2.9.1 General . . . 13

2.9.2 The Design Process . . . 14

2.9.3 Allowable Crack Width . . . 14

2.9.4 Nature and Magnitude of Restraint . . . 15

2.9.5 The Magnitude of Restrained Strain and the Risk of Cracking . . . 15

2.9.6 Crack-inducing Strain . . . 15

2.9.7 Minimum Area of Reinforcement . . . 17

2.9.8 Check the Reinforcement for Crack Control, Crack Spacing and Width . . 17

3 Models for Crack Width Calculation 19 3.1 Calculation of Crack Widths in Structures with Combined Reinforcement . . . . 19

3.2 Shrinkage Cracking in Fully Restrained Members . . . 21

3.2.1 Calculation of Restraining Force and Internal Stresses . . . 21

3.2.2 Calculation of Final Stresses and Deformation . . . 23

3.3 Early-age Thermal Crack Control in Concrete . . . 26 ix

4 Laboratory Experiment 29

5 Construction Site Visit 33

5.1 Sandsg˚ard Bridge . . . 33

5.1.1 Comments . . . 37

5.2 Residual Flexural Tensile Strength . . . 37

6 CrackTeStCOIN 41 6.1 General . . . 41

6.2 Modeling the Laboratory Experiment . . . 41

6.2.1 Simulation specifications . . . 41

6.2.2 Results . . . 42

6.2.3 Comment . . . 45

6.3 Modeling Sandsg˚ard Bridge . . . 46

6.3.1 Simulation specifications . . . 46

6.3.2 Results . . . 46

7 Temperature Calculations - Sandsg˚ard Bridge 51 7.1 Temperature Stresses . . . 51

7.2 Autogenous Shrinkage . . . 55

7.3 Total Concrete Stress . . . 55

8 Calculation of Crack Widths due to Shrinkage Cracking in Fully Restrained Members 57 9 Early-age Thermal Crack Control - Calculation of Crack Widths 61 9.1 Laboratory Experiment . . . 61

9.1.1 Comment . . . 63

9.2 Sandsg˚ard Bridge . . . 63

10 Calculation of Crack Widths in Structures with Combined Reinforcement 67 10.1 The Laboratory Experiment . . . 67

10.2 Sandsg˚ard Bridge . . . 68

10.2.1 Procedure . . . 68

10.2.2 Input Data . . . 68

10.2.3 Results . . . 69

10.3 Modification of the Model . . . 70

10.3.1 Input Data . . . 70

10.3.2 Results . . . 71

11 Discussion 73 11.1 Comparison of the Results . . . 73

11.2 Modeling in CrackTeStCOIN . . . 76

11.3 When do the Cracks Occur? . . . 77

12 Concluding Remarks 79 12.1 Further Work . . . 80

Bibliography 81 A Drawings of Sandsg˚ard Bridge 83 A.1 Reinforcement Drawing . . . 83

A.2 Calculation of the Cross Sectional Area of the Edge Beams . . . 84

A.3 Constructional drawings of Sandsg˚ard Bridge . . . 85

B Residual Flexural Tensile Strength 87

C Temperature Development in Curing Boxes 91

αe, η The modular ratioEs/Ecm

α_T The coefficient of thermal expansion in concrete

∆T Temperature difference

∆σca Concrete stress due to autogenous shrinkage

∆εT The thermal dilation

φ,φ_b The reinforcement bar diameter φ^∗ The final creep coefficient φef The effective creep coefficient ρ The ratioAs/Act

σ_T Concrete stress due to tempeature changes σav The average concrete stress

σc1 Compressive stress in concrete σ_s1 Compressive stress in reinforcement σ_s2 Tensile stress in reinforcement σs The stress in the reinforcement

εca(t) The autogenous shrinkage strain in concrete ε_cr The crack-inducing strain

εctu The tensile strain capacity

εf ree The strain which would occur if the member was completely unrestrained εr Restrained strain

A_I A_c+A_s(^E_E^s

c−1)

xiii

A_n The cross-sectional area of the new (restrained) pour Ao The cross-sectional area of the old (restraining) concrete As The reinforcement area

Act The area of concrete in tension

A_ef The effective concrete area equal to b×h_ef, whereh_ef is the part of the tensile zone which has the same centre of gravity as the reinforcement

As,min The minimum area of reinforcement E_c The modulus of elasticity of the concrete

E_n The modulus of elasticity of the new pour concrete Eo The modulus of elasticity of the old concrete Es The modulus of elasticity of the reinforcement f_t The tensile strength of the concrete

fckt,0.05 70% of the mean axial tensile strength of the concrete fck The compressive strength of the concrete

fcm The average compressive concrete strength

f_{ct,ef f} The mean value of the tensile strength of the concrete effective at the time when the cracks may first be expected to occur

fctm The average tensile strength of the concrete f_{f res} The residual flexural tensile strength

f_R1 The characteristic residual flexural tensile strength at a crack mouth opening displace- ment of 0.5 mm

fyk Characteristic yield strength of the reinforcement

k The coefficient which allows for the effect of non-uniform self-equilibrating stresses, which lead to a reduction of restraint forces

kc A coefficient which takes account of the stress distribution within the section immediately prior to cracking and of the change of the lever arm

l The length of the member n,m The number of cracks

N(σs, ff t,res) The force acting on un-cracked parts of the concrete N₁ The force required to initiate a new crack

Ncr The restraining force in concrete

R The degree of restraint, whereR= 0 for no restraint andR= 1 for full restraint R1 The restraint factor that applies during the early thermal cycle

R2 The restraint factor applying to medium and long-term deformation R3 The restraint factor applying to drying shrinkage

R_j The restraint at the joint

Rax A factor defining the degree of external restraint provided by elements attached to the element considered

R_edge The edge restraint factor at the location of the maximum crack width s₀ The distance over which the concrete and steel stresses vary

sr,max The maximum crack spacing

t The age of the concrete at the given time

T₁ The difference between the peak temperature and the mean ambient temperature at the end of the thermal cycle

T2 The difference between the mean ambient temperature at the end of the early thermal cycle and the minimum element temperature likely in the course of the element life t_s The age of the concrete in days at the beginning of drying.

w/c The water/cement-ratio in concrete.

wk, w The crack width

(ε_sm−ε_cm) The difference in deformation between the steel and the concrete over the maximum crack spacing

αc The coefficient of thermal expansion in concrete ^∗₁ The final concrete strain

^∗_c The final creep strain e The final elastic strain ^∗_s1 The final steel strain ^∗_sh The final shrinkage strain ε_c Creep strain in concrete εsh Shrinkage strain in concrete

Eef f Effective modulus of elasticity of concrete

Chapter 1

Introduction

1.1 Background

This master thesis is written as a continuation of the specialization project written by the un- dersigned. On many of today’s bridges, the edge beams are cast some weeks after the casting of the bridge deck. Temperature differences between two subsequent casting operations give a disadvantageous stress development in the concrete structure. The most commonly used struc- ture to illustrate this is the wall cast on an existing foundation. During heating the newly cast structure, the wall, expands freely, due to its low stiffness. In the cooling phase, however, the wall has developed considerable strength and stiffness, at the same time as the adhesion against the foundation is almost fully developed. The foundation will restrain the contraction in the wall during the cooling. This external restraint will induce tensile stresses in the wall, which in turn may cause cracks, as shown in Figure 1.1. [1]

Figure 1.1: Crack development due to external restraint. Image adapted from Concrete Technology 1, TKT 4215 [1].

Such cracks do often go through the entire thickness of the structure, and may reduce the quality of the structure. For instance, does the high alkalinity of the concrete’s pore solution lead to a formation of a thin, dense film of corrosion products on the reinforcement bar surface. Cracks may lead chloride ions or carbon dioxide gas in to the steel, which will lower the pH in the

1

concrete. This may destroy the thin protective layer, which in turn may lead to corrosion, as shown in Figure 1.2. This may lead to a reduced cross section of the reinforcement, further cracking of the concrete or hydrogen embrittlement. [1]

Figure 1.2: The electrochemical reactions in the corrosion process. Image adapted from Concrete Technology 1, TKT 4215 [1].

Due to the fact that many structural elements in bridges are exposed to external restraint, a laboratory experiment was carried out in connection with the specialization project, intentionally to reduce cracks in edge beams by the use of fiber reinforcement. The experiment studied the effect of steel fibers and polymer fibers on the crack development, compared to a concrete without fiber reinforcement.

The experiment showed that the fiber reinforcement had a very positive effect on the crack development. In the fiber mixes, one saw a much denser crack development than for the reference concrete, and the crack widths were much smaller in the fiber mixes. The steel fibers reduced the average crack width by a factor of 4-5, while the polymer fibers reduced the crack width by a factor of 2-3. This effect is favorable for the concrete, since it reduces the risk of corrosion, and improves the aesthetics of the concrete surface.

A model for calculating crack widths, developed by Ingemar L¨ofgren [2], was used to calculate the expected crack widths in the edge beams. The measured crack widths were compared with the calculated values, shown in Table 1.1. The calculation model is only valid for a member that is fully restrained in both ends, which may explain parts of the differences in the calculated and measured values. In this master thesis the model is studied, and due to the scope of the thesis, the model has been modified to account for the considered edge beams.

Table 1.1: Average measured crack widths and calculated maximum crack widths for the edge beams in the laboratory experiment.

Concrete mix Measured crack width Calculated crack width

(average) (maximum)

Reference mix 0.17 0.09

Concrete with steel fibers 0.03-0.04 0.02

Concrete with polymer fibers 0,06 0.03

Two other calculation models for crack widths are also presented and used in this thesis. These are not taking the effect of fiber reinforcement into account in the calculation, and are therefore only performed on the reference beams.

On the initiative of Norwegian Public Roads Administration, Region West, it was decided to carry out the same measures as in the laboratory experiment, to reduce cracks on a real bridge.

This large-scale experiment has been conducted in connection with this thesis. The chosen bridge is the pedestrian and bicycle bridge Sandsg˚ard Bridge in Ganddal, close to Sandnes. The bridge is 50 meters long and the edge beams were cast with six different concrete mixes, five of them containing fiber reinforcement. Two of the presented calculation models are used to calculate the theoretical crack widths on this bridge.

CrackTeStCOIN is a finite element computer program, that simulates temperatures and stresses in concrete. This program is used to estimate the stress case for both the laboratory experiment and the full-scale experiment. The concrete strain are derived from the estimated stresses, and the modeled and calculated strains are later compared.

Chapter 2

Theory

This chapter includes some relevant theoretical background. First the temperature effects on concrete is discussed, followed by creep and shrinkage in concrete. Then, strains and stresses in concrete subjected to restrained imposed deformations are discussed. Further, measures to reduce effects of cracking is presented. Lastly, a procedure for early-age thermal crack control is presented.

2.1 General

Concrete has a low tensile strength and tensile strain capacity, and the crack development starts already at a tensile strain of about 0.1h. The drying shrinkage of concrete is about 0.6-0.8h, which means that it is almost impossible to avoid cracking. Hence, reinforcement is needed to control the behaviour after cracking and to limit the crack widths. Large crack widths may lead to accelerated reinforcement corrosion in severe environments, leakage in water-retaining/resisting structures, insanitary conditions, or obstructions and interruptions in production processes. [2]

By preventing cracking, the chance that water can lead harmful salts into the reinforcement decreases, and thus decreases the risk of corrosion of the reinforcement. With knowledge of how and why cracks occurs, it is possible to select a concrete composition and execution that ensures durability [3].

2.2 Temperature

The temperature changes that occurs during the hardening phase and due to climate variations, lead to deformation of the concrete. This deformation is called thermal dilation, and is expressed by the equation:

∆εT = ∆T×αT (2.1)

5

where ∆ε_T is the thermal dilation, ∆T is the temperature difference andα_T is the coefficient of thermal expansion.

In a restraint case may such temperature loads lead to thermal cracking. The thermal effect is generally the main contributor to stresses in concrete in the hardening phase [1]. In the Nor- wegian Public Roads Administration’s Prosesskode 2 the following temperature requirements, illustrated in Figure 2.1, have been formulated to limit the amount of early age cracking:

• The maximum temperature shall at no time exceed 65 ^◦C.

• The temperature differential over the cross-section shall not exceed 20 ^◦C

• The difference between the average temperatures in two adjacent cast sections shall not exceed 15^◦C if the restraining length between cast sections exceeds 5 metres.

Figure 2.1: Temperature requirements in accordance with the Norwegian Public Roads Ad- ministration. Figure adapted from Concrete Technology 1 TKT 4215 [1].

2.3 Effect of Temperature on the Material Properties of Concrete

2.3.1 General

This section covers the temperature effects on the material properties strength, stiffness and creep of temperatures in the range -25^◦C to 200^◦C. All literature presented in this section is obtained from Eurocode 2: Design of concrete structures, Part 3: Liquid retaining and containment structures - Annex K [4]. While reading, it is important to keep in mind that the changes are strongly dependent on the particular type of concrete used.

2.3.2 Material Properties at Sub-zero Temperatures

When concrete is cooled below zero, its strength and stiffness increases. The higher the moisture content is, the greater the increase in strength and stiffness is. For saturated concrete, a cooling of the concrete to -25^◦C leads to an increase of around 30 MPa in compressive strength, while the same cooling leads to an increase of around 5 MPa for partially dry concrete. Cooling concrete to - 25^◦C leads to an increase in the modulus of elasticity of around 8000 MPa for saturated concrete and 2000 MPa for partially dry concrete.

2.3.3 Material Properties at Elevated Temperatures

Youngs modulus may be assumed to be unaffected by temperature up to 50^◦C. For higher temperatures, a linear reduction may be assumed up to a reduction of 20 % at a temperature of 200^◦C.

The creep coefficient may be assumed to increase with increasing temperature above 20^◦C for concrete heated prior to loading. Table 2.1 presents the appropriate creep coefficient multipliers.

Table 2.1: Creep coefficient multiplier for different temperatures.

Temperature (^◦C) Creep coefficient multiplier

20 1.00

50 1.35

100 1.96

150 2.58

200 3.20

If the load is present during heating of the concrete, this will lead to excess deformations that are irrecoverable.

2.4 Creep and Shrinkage of Concrete

Normally there will exist quite large stresses between the constituents in concrete, due to the inhomogeneous structure. This stress case arises partly because of the differences in the elastic and thermal properties between the cement paste and the aggregates. Another important reason is the tendency to volume reduction in the cement paste during drying.

When concrete is exposed to external forces, the internal and external forces will superimpose.

This leads to a stress case where the elements are partly less or larger than in the stress case for the external stresses. Creep is time dependent deformation in a material, due to external loading [5]. Shrinkage is change in volume, due to change in the concretes moisture content [6].

If inconsistent drying is the case, the cement paste will be in tension and the coarse aggregates in compression. A stress case like this may explain the many conditions which occur during deformation of concrete [7].

A study of the specific effect of periodic variation in stresses, showed that a characteristic stress applied as a constant stress equalised the varying stresses within the reasonable limits, so far as the development of creep was concerned. Periodic changes in moisture conditions during the first months of loading influenced the creep, as well as the shrinkage. It was found that immersion in water for short periods of time also reduces the long-time values of creep and shrinkage [7].

2.5 Autogenous Shrinkage

The autogenous shrinkage is the self-produced shrinkage of the concrete. When cement and water reacts, the reaction product fills a smaller volume than the reactants, and chemical shrinkage occurs. At full hydration for a cement paste with w/c = 0.40, a volume loss of approximately 8 % is estimated. This volume loss creates chemical shrinkage pores. During further hydration these pores are partly emptied, which leads to a decrease in relative humidity within the concrete. This phenomena is called self-desiccation. The process creates capillary forces and under-pressure in the pore water. This results in an external contraction of the concrete, which is called autogenous shrinkage. For high strength concrete it is not unusual that the autogenous shrinkage is 0.1 - 0.2 h, while the tensile strain capacity of concrete generally is around 0.1 h. Naturally this will give rise to problems in practice. [1]

Most of the autogenous shrinkage strain develops in the early age of the concrete. According to Eurocode 2, Part 1-1, 3.1.4 (6) the autogenous shrinkage strain is given by the equation:

εca(t) =βas(t)εca(∞) = (1−e^−0.2t0.5)×2.5(fck−10)×10⁻⁶ (2.2)

[8]

2.6 Strains and Stresses in Concrete Sections Subjected to Restrained Imposed Deformations

All literature presented in this section is obtained from Eurocode 2: Design of concrete structures, Part 3: Liquid retaining and containment structures - Annex L [4].

The strain in an uncracked concrete section is calculated from factors defining the degree of both axial and moment restraint. These factors are dependent on the stiffness of the element considered, and the geometry of the members attached to it. Restraint factors for a wall on a base is shown in Figure 2.2, and Table 2.2 presents the restraint factors for a central zone in a wall.

Figure 2.2: Restraint factors for wall on base. Figure adapted from Eurocode 2, Part 3, Annex L [4].

Table 2.2: Restraint factors in central zone.

Ratio L/H Restraint factor at base Restraint factor at top

1 0.5 0

2 0.5 0

3 0.5 0.05

4 0.5 0.3

>8 0.5 0.5

The stresses in concrete are calculated from the strains, and are consequently also dependent of the restraint.

2.7 Calculation of Crack Widths due to Restraint of Im- posed Deformations

All literature presented in this section is obtained from Eurocode 2: Design of concrete structures, Part 3: Liquid retaining and containment structures - Annex M [4].

2.7.1 General

This section covers shrinkage and the early thermal movement due to cooling of members right after casting. Two practical problems, related to different forms of restraint, are investigated.

The restraint along a members ends and restraint along one edge are studied, illustrated in Figure 11.1. The first-mentioned occurs when a new section of concrete is cast between two pre-existing sections, while the second case arises where a wall is cast onto a pre-existing stiff base.

(a) Restraint of a member at its end. (b) Restraint along one edge.

Figure 2.3: Types of restraint to walls. Figure adapted from Eurocode 2, Part 3, Annex M [4].

2.7.2 Restraint of a Member at its End

The maximum crack width may be calculated from the expression:

wk=sr,max×(εsm−εcm) (2.3)

where sr,max is the maximum crack spacing and (εsm−εcm) is the difference in deformation between steel and concrete over the maximum crack spacing.

For a member restrained at its end (ε_sm−ε_cm) may be calculated from the following expression:

sm−cm=0.5αekckfct,ef f

Es

(1 + 1

αeρ) (2.4)

where

• α_eis the ratioE_s/E_cm

• kc is a coefficient which takes account of the stress distribution within the section imme- diately prior to cracking and of the change of the lever arm.

• k is the coefficient which allows for the effect of non-uniform self-equilibrating stresses, which lead to a reduction of restraint forces.

• fct,ef f is the mean value of the tensile strength of the concrete effective at the time when the cracks may first be expected to occur.

• ρ is the ratioA_s/A_ct, whereA_sis the reinforcement area and A_ct is the area of concrete in tension.

One may check cracking without direct calculation. This is done by calculating σs from the expression:

σs= kckfct,ef f

ρ (2.5)

2.7.3 Restraint Along One Edge

In case of restraint along one edge, the formation of a crack only influences the distribution of stresses locally, and the crack width is therefore a function of the restrained strain. The crack width may be estimated by the expression:

(εsm−εcm) =Raxεf ree (2.6)

whereR_ax is a factor defining the degree of external restraint provided by elements attached to the element considered andεf ree is the strain which would occur if the member was completely unrestrained.

Figure 2.4 illustrates the difference between cracking in case of end and edge restraint.

2.8 Measures to Reduce Effects of Cracking

In this section it is discussed how to design concrete structures to reduce effects of cracking due to restraint of imposed deformations from thermal effects and shrinkage. All literature is obtained from a note written by Terje Kanstad, prepared as basis for a meeting in CEN WG1/TG7 Karlsruhe February 1 and 2, 2016: Draft for Annex D for Eurocode 2 - 2020: Guidance to restrict early age cracking + Quotation of relevant crack width equations (Revised February 26th 2016).

The most serious problem concerning both durability and serviceability is through-cracking.

The amount of these cracks may, however, be reduced by performing some measures. Possible measures, presented in the draft, are to use low heat concretes, concretes with a low coefficient of thermal expansion, heating cables in the restraining structural elements, cooling pipes in the hardening concrete, reduced fresh concrete temperature, or finally to reduce the degree of restraint for the hardening concrete structural element.

Figure 2.4: Relation between crack width and imposed strain for edge and end restrained walls. Figure adapted from Eurocode 2, Part 3, Annex M [4].

Figure 2.5(a) illustrates the expansion phase corresponding to the heating phase, while Figure 2.5(b) shows the contraction phase which is an effect of the cooling. Both figures also illustrate the typical distribution of stresses over the wall height. The most critical time for cracking,t_crit, is usually assumed to be when the wall temperature reaches the temperature of the surroundings.

(a) Concrete wall in expansion phase. (b) Concrete wall in contraction phase.

Figure 2.5: Illustration of the expansion and contraction phase in a wall, as well as the stress distribution. Figure adapted fromDraft for Annex D for Eurocode 2 -2020.

Related to temperature development in the hardening concrete, the most decisive parameters are the fresh concrete temperature,T_ci, and the ambient temperature history. In addition, also the insulation conditions of the surface of the member, the wind conditions and the solar radiation influence the concrete temperature history. To describe temperature history, the fresh concrete temperature, the maximum concrete temperature and the temperature of the surroundings have

to be considered. To obtain a complete crack assessment, also the minimum temperatures due to the seasonal variations must be determined, both for exposed and restraining structures.

Reinforcement may reduce crack spacing and limit the crack widths, but it cannot, however, prevent early age cracking. The imposed load effects related to early age cracking are often membrane actions, as for through cracking, or simply local effects, such as surface cracking.

These differ from the load effects presumed in the minimum reinforcement rules, and are hence a complicated problem for designers. The minimum reinforcement rules generally relate to cross section behavior with linear strain distribution, which in general does not hold in massive concrete structures, where early age cracking problems may occur.

It has been assumed that limiting the temperature differences across the cross-section of a mem- ber to 20^◦C, or between a hardening and a restraining member to 15^◦C, will give a structure without early age cracking. These are very rough requirements, and they are solely connected to temperature and obviously insufficient on a fundamental level since the other parameters involved are not considered.

Considering prevention or limitation of cracking due to imposed deformations by strain and stress calculations, the methods may be grouped into three levels:

(I) Simple conservative methods with default values

(II) Reasonably simple methods with representative input parameters

(III) Advanced timedependent FE Analyses for temperatures, strains and stresses.

2.9 Early-age Thermal Crack Control in Concrete

2.9.1 General

In this section a procedure for controlling early-age thermal cracking is presented. The theory is obtained from the British guidelineEarly-age thermal crack control in concrete - CIRIA C660 [9]. During hydration of concrete, heat is released from the clinker, which may cause early-age thermal cracking. When the tensile strain, arising from either restrained thermal contraction or a temperature differential within the concrete, exceeds the tensile strain capacity, early-age thermal cracking occurs. Autogenous shrinkage may also contribute to early contraction. In thin sections, early-age thermal cracking may occur within a few days. Some of the factors that influence the risk of early-age thermal cracking are listed below.

• Temperature rise

• The coefficient of thermal expansion of the concrete

• The restraint to movement offered by adjacent elements

• The restraint to movement offered by differential strain within an element

• The ability of the concrete to resist tensile strain

External restraint may be in two principal forms, either continuous edge restraint or end restraint.

In the case with edge beams on a bridge, it is edge restraint that is the restraint case.

The temperature rise in the concrete depends on the capacity of the concrete to generate heat, depending on content and type of cement, the element thickness and the curing conditions.

2.9.2 The Design Process

The steps involved in the design process for assessing and controlling early age cracking are listed below.

1. Define the allowable crack width

2. Define the nature and magnitude of restraint

3. Estimate the magnitude of restrained strain and the risk of cracking 4. Estimate the crack-inducing strain

5. Check the minimum area of reinforcementAs,min

6. Check the reinforcement for crack control, crack spacing and width

2.9.3 Allowable Crack Width

It is not common practice to add early-age crack widths to the crack widths arising from struc- tural loading. Nevertheless, the designer should consider whether or not to add cracking due to subsequent deformations to the early-age effects. Depending on the restraint, long-term thermal contractions and drying shrinkage may either cause crack widths to increase or new cracks to form.

If a member is subjected to continuous edge restraint and the restraint is maintained, one may expect the cracks that are formed at a early age to widen. If both the retained and the restraining member are exposed to the same environment, only the differential deformation needs to be considered when estimating the crack widening.

In members subjected to end restraint, the crack width is determined by the strength of the concrete at the time of cracking. As the strength develops, the restrained contraction will increase the existing crack widths and cause new cracks to form. These cracks are wider than the early-age cracks.

The crack width is given by the expression:

w_k=S_r,max(ε_sm−ε_cm)

where Sr,maxis the maximum crack spacing, as shown in Figure 2.6, εsm is the mean strain in the reinforcement andεcmis the mean strain in the concrete between cracks.

Figure 2.6: Illustration of maximum crack spacing in reinforced concrete.

2.9.4 Nature and Magnitude of Restraint

The restraint plays an important role in determining the restrained strain and risk of cracking.

If the restraint is low, early thermal cracking may be avoided, and hence also the early thermal cracking control. If cracking is predicted, the nature of restraint (end or edge) has a significant impact on the amount of reinforcement needed.

In some cases the restraining members may be subject to a greater risk of cracking than the newly cast member.

2.9.5 The Magnitude of Restrained Strain and the Risk of Cracking

The risk of cracking should be evaluated in two different cases: Both when early temperature change and autogenous shrinkage will be the major cause of strain during the early period, and later when temperature changes and drying shrinkage will dominate.

If the cracking that has developed in the early period is unacceptable, one must consider the options for reducing the restraint and/or the thermal strain.

If the risk of cracking is acceptable low, then there is no need to design or to check reinforcement specifically to control cracking caused by restrained contraction.

2.9.6 Crack-inducing Strain

Eurocode 2, Part 3 defines the crack-inducing strain as the difference between the mean strain in the reinforcement and the mean strain in the concrete after a crack has occurred [4]. In case

of restraint of a member at its ends, as shown in Figure 2.7(a), the crack-inducing strain is given by the expression:

εsm−εcm=0.5αekckfct,ef f

E_s

1 + 1

α_eρ

(2.7)

where

• k is a coefficient which allows for the effect of non-uniform and self-equilibrating stress which leads to a reduction in restraint forces

• k_c is a coefficient which takes account of the stress distribution within the section imme- diately prior to cracking

• fct,ef f is the mean tensile strength of the concrete at the time of cracking

• E_s is the modulus of elasticity of the reinforcement

• α_eis the modular ratio

• ρis the ratioAs/Act

• Asis the total area of reinforcement

• A_ctis the gross section in tension

(a) End restraint (b) Edge restraint

Figure 2.7: The difference between edge and end restraint. Figure adapted fromEarly-age thermal crack control in concrete[9].

In case of edge restraint, illustrated in Figure 2.7(b), the crack-inducing strain is given by the expression:

εsm−εcm=Raxεf ree (2.8)

where

• Rax is the restraint factor

• ε_{f ree} is the strain which would occur if the member was completely unrestrained

2.9.7 Minimum Area of Reinforcement

Eurocode 2, Part 1-1 presents an expression for the minimum area of reinforcement, which ensures that the reinforcement doesn’t yield under the load transferred from the concrete to the steel when a crack occurs [8]. For members subjected to continuous edge restraint, this expression is over-conservative. It has therefore been modified for this case:

A_s,min=k_Redgek_cA_ctfckt,0.05(t)

f_yk (2.9)

wherefckt,0.05is 70% of the mean axial tensile strength ,fyk is the characteristic yield strength of the reinforcement, andkRedgeis conservatively calculated from the expression:

kRedge= 1−0.5×Redge (2.10)

whereR_edgeis the edge restraint factor at the location of the maximum crack width.

2.9.8 Check the Reinforcement for Crack Control, Crack Spacing and Width

If the estimated crack width exceeds the upper limit, one may consider options to reduce the extent of cracking, for example by increasing the amount of reinforcement.

If the elements are subjected to edge restraint, the crack width is strain limited and may be reduced by achieving less restrained thermal strain. For the end restraint case, on the other hand, a decrease in thermal strain will only reduce the number of cracks, not the crack width.

In this case one should rather decrease the restraint or increase the amount of reinforcement.

[9]

Chapter 3

Models for Crack Width Calculation

In this chapter three different methods for calculating the crack width in concrete are presented.

First a calculation method for crack widths in structures with combined reinforcement is pre- sented. That is structures containing both ordinary reinforcement bars and fiber reinforcement.

The second model that is presented is a calculation model for crack widths in structures exposed to shrinkage cracking in fully restrained members. Lastly, a method for early-age thermal crack control is presented.

3.1 Calculation of Crack Widths in Structures with Com- bined Reinforcement

This section is influenced by the paper presented at Nordic Mini-seminar ”Fiber reinforced concrete”,Calculation of crack width and crack spacing, written by Ingemar L¨ofgren [2].

L¨ofgren states that almost no guidelines exist for structural engineers concerning structures having both fibre- and bar reinforcement. If cracking is caused by an imposed deformation, the force in the member depends on the actual stiffness and the crack width on the number of cracked formed. Engstr¨om [10] has proposed a model for analysing restraint induced cracking in concrete with ordinary bar reinforcement. The cracking process is analysed by modeling the reinforcement in the cracks as non-linear springs, illustrated in Figure 3.1. L¨ofgren has extended the model to also include the effect of fibre reinforcement. This model is presented in this section.

L¨ofgren’s model shows how the combined effect of bar reinforcement and fibre bridging influences the crack spacing and width in the serviceability limit state. Figure 3.2 illustrates how forces are acting on un-cracked parts of concrete with both fiber reinforcement and ordinary reinforcement bars. The model is only valid for cracking caused by restraint stresses. It is based on a bond-slip relationship between reinforcement and concrete, as well as a compatibility requirement for the combined material.

19

Figure 3.1: The reinforcement in the cracks modelled as non-linear springs [2].

Figure 3.2: Forces acting on an un-cracked concrete with combined reinforcement[2].

Based on a bond-slip relationship an analytical expression has been derived, describing the crack width as a function of the reinforcement stress

w(σs) = 0.42 φ×σ_s² 0.22×f_cmE_s

1 +^E_E^s

c ×_A^As

ef

!

(3.1)

where

• φis the bar diameter

• σsis the stress in the reinforcement

• f_cmis the average compressive concrete strength

• E_s andE_c is the modulus of elasticity of the reinforcement respectively the concrete

• Aef is the effective concrete area

The effective concrete area is calculated asAef =b×hef , wherehef is the part of the tensile zone which has the same centre of gravity as the reinforcement.

The response during the cracking process may be described with the following deformation criteria:

N(σs, ff t,res)×l

E_c×A_I (1 +φef) +n×w(σs) =R×cs×l (3.2)

where

• N(σ_s, f_{f t,res}) is the force acting on un-cracked parts

• l is the length of the member

• A_I =A_c+A_s(^E_E^s

c −1)

• φef is the effective creep coefficient

• n is the number of cracks

• R is the degree of restraint, whereR= 0 for no restraint and R= 1 for full restraint N(σs, ff t,res) may be calculated as:

N(σs, ff t,res) =σs×As+ff t,res(Aef −As) (3.3) IfN(σ_s, f_{f t,res}) is larger than the force required to initiate a new crack,N₁, more cracks will be formed. However, if it is smaller only one crack will be formed. The force required to initiate a new crack,N1, can be calculated as:

N₁=f_ctm

A_ef +Es

E_c −1

(3.4) wherefctm is the average tensile strength.

IfN(σs, ff t,res)> N1a new crack is initiated andnincreases. IfN(σs, ff t,res)< N1the cracking process stops and the actual crack width can be determined using equation 3.1.

[2]

3.2 Shrinkage Cracking in Fully Restrained Members

In this section the problem of cracking in fully restrained members subjected to direct tension caused by drying shrinkage is considered. A rational approach for determination of number and spacing of cracks, as well as the average crack width are presented. The approach is obtained from ACI Structural Journal, Vol.89 no.2 [11].

3.2.1 Calculation of Restraining Force and Internal Stresses

To determine the crack width w and the stresses, the distances0 has to be known. s0 is the distance over which the concrete and steel stresses vary. Further on, the restraining force,Ncr, has to be calculated. The stress in the concrete varies from zero at the crack, to compressive stress,σc1atx=s0. The stress in the steel varies from the tensile stress,σs2, to the compressive stress,σs1, atx=s0.

An approximation fors₀ may be obtained from the following equation:

s0= db

10ρ (3.5)

wheredb is the bar diameter andρis the reinforcement ratio As/Ac.

The following procedure shows how to calculate N_cr immediately after the first cracking, as well as the corresponding stresses. Since the member is fully restrained, it is prevented from shortening, and hence the overall elongation of the steel is zero. Integrating the steel strain over the length of the member gives:

εs1

Es

L+σs2−σs1

Es

2

3s0+w

= 0 (3.6)

Sincew is much less thans0,w can be neglected. A rearrange of Equation 3.6 gives:

σs1= −2s0

3L−2s₀σs2 (3.7)

At the crack the restraining force is carried entirely by the steel, which gives the equation for the stress at the crack as:

σs2=Ncr

As

(3.8) The steel stress away from the crack is given by substituting Equation 3.8 into Equation 3.7:

σs1= 2s0

3L−2s₀ Ncr

A_s =−C1

Ncr

A_s (3.9)

where

C₁=− 2s0

3L−2s₀ (3.10)

Because the member is fully restrained, the total concrete strain is zero at any point prior to cracking. The creep and elastic strains are tensile and the shrinkage strain is compressive (negative). Immediately before the first crack occurs, the sum of the creep and the shrinkage strain components is

ε_c+ε_sh=−f_t Ec

(3.11)

whereftis the concrete stress andEc is the elastic modulus of the concrete at the time of first cracking. Immediately after first cracking, the magnitude of the elastic component of strain in the uncracked concrete decreases, but the creep and shrinkage strain components are unaltered.

At any distance greater thans_o from the crack, equilibrium requires that the sum of the forces in the concrete and the steel immediately after first cracking is equal toNcr.

σc1Ac+σs1As=Ncr (3.12)

Substituting Equation 3.9 into Equation 3.12 and rearranging gives

σ_c1= N_cr−σ_s1A_s Ac

= N_cr(1 +C₁) Ac

(3.13) The compatibility requirement is that the concrete and steel strains at any distance greater than s0 from the crack, are identical. That is

s1=1 (3.14)

which can be expressed as

σ_s1 Es

= σ_c1 Ec

+_c+_sh (3.15)

Substituting Equation 3.9, 3.11 and 3.13 into Equation 3.15 and solving forN_cr gives

Ncr= ηρftAc

C1+ηρ(1 +C1) (3.16)

whereρ= ^A_A^s

c andη = ^E_E^s

c. WhenNcr is calculated, the concrete and steel stresses immediately after cracking may be calculated from from Equation 3.8, 3.9 and 3.13.

3.2.2 Calculation of Final Stresses and Deformation

The final stresses and deformation are indicated with an asterisk(*). By equating the overall elongation of the steel to zero, the following expression for a member containing m cracks is obtained:

σ^∗_s1

E_sL+mσ^∗_s2−σ_s1^∗ E_s (2

3s₀+w) = 0 (3.17)

The crack widthw is much less thans0, and can therefore be neglected. Rearranging the above equation gives the following expression

σ^∗_s1= −2s0m

3L−2s0mσ_s2^∗ (3.18)

Letting the crack spacing s = _m^L gives

σ_s1^∗ = −2s0

3s−2s0

σ_s2^∗ =−C2σ_s2^∗ (3.19)

where

C2= 2s0

3s−2s₀ (3.20)

The final tensile stress in the reinforcement at each crack is:

σ_s2^∗ =N(∞)

A_s (3.21)

The concrete stress history is continuously changing, but it is yet reasonable to assume that the average concrete stress for the estimation of creep strain at any time after the commencement of drying, σav, is somewhere betweenσc1 andft. An approximation of the final creep strain is given by:

^∗_c =σ_av Ec

φ^∗ (3.22)

whereφ^∗ is the final creep coefficient. It is assumed that

σav =σc1+ft

2 (3.23)

At any distance greater thansofrom the crack, the final concrete strain is the sum of the elastic, creep and shrinkage components and may be approximated by:

^∗₁=_e+^∗_c+^∗_sh= σ_av Ec

+σ_av Ec

φ^∗+^∗_sh= σ_av

E_e^∗ +_sh (3.24)

where

E_e^∗= E_c

1 +φ^∗ (3.25)

The final creep coefficient depends on the age at the commencement of drying and the quality of the concrete, and is normally between 2 and 4. ^∗_shis the final shrinkage strain and depends on the relative humidity, the size and shape of the member and the characteristics of the concrete mix.

At any distance greater thanso from the crack, equilibrium requires that the sum of the force in the concrete and the force in the steel is equal toN(∞). That is:

σ^∗_c1Ac+σ_s1^∗As=N(∞)→σ^∗_c1= N(∞)−σ^∗_s1As

Ac

(3.26)

Compatibility gives that the concrete and steel strains are identical:

^∗_s1=^∗₁ (3.27)

This gives:

σ_s1^∗ E_s =σav

E_c^∗ +sh (3.28)

Substituting Equation 3.19 and 3.21 into Equation 3.28 gives:

N(∞) =−n^∗As

C₂ (σav+^∗_shE_e^∗) (3.29) wheren^∗=Es/E_e^∗.

The crack spacing must be determined to calculateC2. σc1must be less than the tensile strength ft. Substituting Equation 3.19 and 3.21 into Equation 3.26, combined with the tensile strength criteria gives:

σ^∗_c1=N(∞)(1 +C2)

A_c 6ft (3.30)

Substituting Equation 3.20 and 3.29 into Equation 3.30 gives:

s6 2s_o(1 +ξ)

3ξ (3.31)

where

ξ= −n^∗ρ(σav+^∗_shE_e^∗)

n^∗ρ(σ_av+^∗_shE_e^∗) +f_t (3.32) The number of cracksm may be taken as the smallest integer that satisfies equation 3.31. Then the restraining force can be calculated using equation 3.29. The steel and the concrete stresses may be determined from equation 3.21.

The overall shortening of the concrete is an estimate of the sum of the crack widths. The final concrete strain at any distance less thans_ofrom the crack is:

^∗₂=f nσ^∗_c1

E_e^∗ +^∗_sh (3.33)

If a parabolic variation of stress in the region less than a distance s_o away from the crack is assumed, the average crack width w is obtained by integrating the concrete strain over the length of the member.

w=− σ_c1^∗

E_e^∗(s−2

3s0) +^∗_shs

(3.34)

If the area of steel, As is small, yielding may occur at each crack and the value of N(∞) will not be correct. In such caseN(∞) is equal tof_yA_s. The stress in the concrete at any distance greater thansofrom the crack is then:

σ_c1^∗ = fyAs−σ^∗_s1As

A_c (3.35)

After the steel at the first crack yields, the tensile stressσc1increases only slightly in the concrete, as the compressive steel stress σ_s1 increases with time and the first crack opens. The width of the crack is usually unacceptably large as the steel at the crack deforms plastically. The crack widthw may be found by insuring that the overall elongation of the steel is zero. That is:

σ^∗_s1 Es

(L−w) +f_y−σ_s1^∗ Es

×2

3s₀+w= 0 (3.36)

Sincew is much smaller thanL, equation 3.36 can be arranged to:

w=−σ_s1^∗(3L−2s0) + 2s0fy

3Es

(3.37) Since the tensile stress in the uncracked concrete does not change significantly with time, it is reasonable to assume that the average concrete stress, σav, is given by equation 3.35 and that the final steel stress at any distance greater than s_o from the first crack may be obtained by substituting equation 3.35 into equation 3.24 and simplifying:

σ^∗_s1

E_s = fsyAs−σ_s1^∗ As

A_cE_e^∗ →σ^∗_s1= n^∗ρfsy+^∗_shEs

1 +n^∗ρ (3.38)

3.3 Early-age Thermal Crack Control in Concrete

The calculation model presented in this section is obtained from the British guideline Early- age thermal crack control in concrete - CIRIA C660 [9]. Extensive theory about the model is presented in Section 2.9.

The allowable crack width is determined from the requirement of a durable concrete structure.

The limiting total crack width arising from early-age deformations, long-term deformations and loading, is 0.3 mm. The full crack pattern is expected to occur at early age under conditions of edge restraint.

To calculate the edge restraint, the restraint at the joint is first estimated,R_j, according to the equation:

Rj= 1 1 +^A_Aⁿ

0 ×^E_Eⁿ

0

(3.39)

where

• An is the cross-sectional area of the new (restrained) pour

• Aois the cross-sectional area of the old (restraining) concrete

• En is the modulus of elasticity of the new pour concrete

• Eo is the modulus of elasticity of the old concrete

R1 is the restraint factor that applies during the early thermal cycle. R2 andR3 are restraint factors applying to medium and long-term deformation and drying shrinkage respectively.

In the early thermal cycle, the modulus of elasticity will be less developed in the newly cast concrete, compared to the old concrete. It therefore is recommended to use a ratio ofE0/En= 0.7−0.8 in the early age.

Restrained strain is expressed as

εr=Kc1[αcT1+ε_ca(3)]R1+Kc2[(ε_ca(28)−ε_ca(3))R2+αcT2R3+εcdR3] (3.40)

whereKc1= 0.65 andKc2= 0.5.

T1is the difference between the peak temperature,Tp, and the mean ambient temperature,Ta, at the end of the thermal cycle.

To predict T1, the cement content is required. While a reduction in the heat generation of the binder is likely to be beneficial, it is the temperature rise in the resulting concrete that is of prin- cipal concern with regard to early thermal cracking. When additions are used, different binder contents are often required to achieve the same strength class of concrete and it is important that this is taken into account when assessing the benefits or otherwise of a particular mix design.

Values ofT1 for CEM II for walls cooling from both faces are given in Figure 9.1. The British guideline presents a table where the total binder content is given for different strength classes, Table 4.2.

T2 is the difference between the mean ambient temperature at the end of the early thermal cycle and the minimum element temperature likely in the course of the element life. For annual temperature changes recommended values of T2 are 20^◦C for concrete cast in the summer and 10^◦C for concrete cast in the winter.

The autogenous shrinkage is obtained from Table 4.5, presented in the British guideline, while the drying shrinkage is calculated from Eurocode 2, 3.1.4 (6).

ts is the age of the concrete in days at the beginning of drying. This is normally at the end of the curing period. tis the age of the concrete at the given time.